Abstract

The theory for linear deformations of fluid microparticles in a laser beam of Gaussian profile is presented, when the beam focus is at the particle center as in optical trapping. Three different fluid systems are considered: water microdroplet in air, air microbubble in water, and a special oil-emulsion in water system used in experiments with optical deformation of fluid interfaces. We compare interface deformations of the three systems when illuminated by wide (compared to particle radius) and narrow laser beams and analyze differences. Deformations of droplets are radically different from bubbles under otherwise identical conditions, due to the opposite lensing effect (converging and diverging, respectively) of the two; a droplet is deformed far more than a bubble, cetera paribus. Optical contrast is found to be of great importance to the shape obtained when comparing the relatively low-contrast oil-emulsion system to that of water droplets. We finally analyze the dynamics of particle motion when the laser beam is turned on, and compare a static beam to the case of a short pulse. The very different surface tension coefficient implies a very different time scale for dynamics: microseconds for the water–air interface and tens of milliseconds for the oil-emulsion. Surface oscillations of a water microdroplet are found always to be underdamped, while those of the oil-emulsion are overdamped; deformations of a microbubble can be either, depending on physical parameters.

© 2013 Optical Society of America

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    [CrossRef]
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  6. C. N. Baroud, M. Robert de Saint Vincent, and J.-P. Delville, “An optical toolbox for total control of droplet microfluidics,” Lab on a Chip 7, 1029–1033 (2007).
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  7. T. Nishimura, Y. Ogura, and J. Tanida, “Optofluidic DNA computation based on optically manipulated microdroplets,” Microfluid. Nanofluid. 13, 1–7 (2012).
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    [CrossRef]
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  47. M. L. Cordero, E. Verneuil, F. Gallaire, and C. N. Baroud, “Time-resolved temperature rise in a thin liquid film due to laser absorption,” Phys. Rev. E 79, 011201 (2009).
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  50. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  51. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  52. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  53. R. C. Thorne, “The asymptotic expansion of Legendre functions of large degree and order,” Phil. Trans. R. Soc. A 249, 597–620 (1957).
    [CrossRef]
  54. S. Å. Ellingsen and I. Brevik, “Electrostrictive fluid pressure from a laser beam,” Phys. Fluids 23, 096101 (2011).
    [CrossRef]
  55. J. V. Wehausen and E. V. Laitone, “Surface waves,” Encyclopedia Phys. 9, 446–476 (1960).

2012

S. Å. Ellingsen, “Microdroplet oscillations during optical pulling,” Phys. Fluids 24, 022002 (2012).
[CrossRef]

T. Nishimura, Y. Ogura, and J. Tanida, “Optofluidic DNA computation based on optically manipulated microdroplets,” Microfluid. Nanofluid. 13, 1–7 (2012).
[CrossRef]

S. Å. Ellingsen and I. Brevik, “Electrostrictive counterforce on fluid microdroplet in short laser pulse,” Opt. Lett. 37, 1928–1930 (2012).
[CrossRef]

2011

A. Schaap, Y. Bellouard, and T. Rohrlack, “Optofluidic lab-on-a-chip for rapid algae population screening,” Biomed. Opt. Express 2, 658–664 (2011).
[CrossRef]

S. Å. Ellingsen and I. Brevik, “Electrostrictive fluid pressure from a laser beam,” Phys. Fluids 23, 096101 (2011).
[CrossRef]

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5, 591–597 (2011).
[CrossRef]

D. Erickson, D. Sinton, and D. Psaltis, “Optofluidics for energy applications,” Nat. Photonics 5, 583–590 (2011).
[CrossRef]

S. Anand, A. Engelbrecht, and D. McGloin, “Optically written optofluidic ice channels,” J. Opt. 13, 044005 (2011).
[CrossRef]

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011), and supplementary information.
[CrossRef]

R. Wunenburger, B. Issenmann, E. Brasselet, C. Loussert, V. Hourtane, and J.-P. Delville, “Fluid flows driven by light scattering,” J. Fluid Mech. 666, 273–307 (2011).
[CrossRef]

B. Issenmann, R. Wunenburger, H. Chrabi, M. Gandil, and J.-P. Delville, “Unsteady deformations of a free liquid surface caused by radiation pressure,” J. Fluid Mech. 682, 460–490 (2011).
[CrossRef]

D. A. Woods, C. D. Mellor, J. M. Taylor, C. D. Bain, and A. Ward, “Nanofluidic networks created and controlled by light,” Soft Matter 7, 2517–2520 (2011).
[CrossRef]

2010

H. Chrabi, D. Lasseux, R. Wunenburger, E. Arquis, and J.-P. Delville, “Optohydrodynamics of soft fluid interfaces: optical and viscous nonlinear effects,” Eur. Phys. J. E 32, 43–52 (2010).
[CrossRef]

2009

P. C. F. Møller and L. B. Oddershede, “Quantification of droplet deformation by electromagnetic trapping,” Europhys. Lett. 88, 48005 (2009).
[CrossRef]

M. L. Cordero, E. Verneuil, F. Gallaire, and C. N. Baroud, “Time-resolved temperature rise in a thin liquid film due to laser absorption,” Phys. Rev. E 79, 011201 (2009).
[CrossRef]

2008

C. Monat, P. Domachuk, C. Grillet, M. Collins, B. J. Eggleton, M. Cronin-Golomb, S. Mutzenich, T. Mahmud, G. Rosengarten, and A. Mitchell, “Optofluidics: a novel generation of reconfigurable and adaptive compact architectures,” Microfluid. Nanofluid. 4, 81–95 (2008).
[CrossRef]

Z. Li and D. Psaltis, “Optofluidic dye lasers,” Microfluid. Nanofluid. 4, 145–158 (2008).
[CrossRef]

O. J. Birkeland and I. Brevik, “Nonlinear laser-induced deformations of liquid–liquid interfaces: an optical fiber model,” Phys. Rev. E 78, 066314 (2008).

2007

R. D. Schroll, R. Wunenburger, and J.-P. Delville, “Liquid transport due to light scattering” Phys. Rev. Lett. 98, 133601 (2007).

C. N. Baroud, M. Robert de Saint Vincent, and J.-P. Delville, “An optical toolbox for total control of droplet microfluidics,” Lab on a Chip 7, 1029–1033 (2007).
[CrossRef]

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: a new river of light,” Nat. Photonics 1, 106–114 (2007).
[CrossRef]

2006

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442, 381–386 (2006).
[CrossRef]

A. D. Ward, M. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. 2006, 4515–4517 (2006).
[CrossRef]

2005

A. Hallanger, I. Brevik, S. Haaland, and R. Sollie, “Nonlinear deformations of liquid–liquid interfaces induced by electromagnetic radiation,” Phys. Rev. E 71, 056601 (2005).
[CrossRef]

2003

A. Casner and J.-P. Delville, “Laser-induced hydrodynamic instability of fluid interfaces,” Phys. Rev. Lett. 90, 144503 (2003).

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
[CrossRef]

2002

S. Mitani and K. Sakai, “Measurement of ultralow interfacial tension with a laser interface manipulation technique,” Phys. Rev. E 66, 031604 (2002).
[CrossRef]

2001

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767–784 (2001).
[CrossRef]

2000

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef]

1999

1995

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef]

1991

1989

H.-M. Lai, P. T. Leung, K. L. Poon, and K. Young, “Electrostrictive distortion of a micrometer-sized droplet by a laser pulse,” J. Opt. Soc. Am. B 6, 2430–2437 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1988

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J.-Z. Zhang and R. K. Chang, “Shape distortion of a single water droplet by laser-induced electrostriction,” Opt. Lett. 13, 916–918 (1988).
[CrossRef]

1986

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–431 (1986).
[CrossRef]

1981

H. M. Lai, W. M. Suen, and K. Young, “Microscopic derivation of the Helmholtz force density,” Phys. Rev. Lett. 47177–179 (1981).
[CrossRef]

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
[CrossRef]

1978

R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. R. Soc. London Ser. A 360, 347–363 (1978).
[CrossRef]

1976

R. Peierls, “The momentum of light in a refracting medium,” Proc. R. Soc. London Ser. A 347, 475–491 (1976).
[CrossRef]

H.-M. Lai and K. Young, “Response of a liquid surface to the passage of an intense laser pulse,” Phys. Rev. A 14, 2329–2333 (1976).
[CrossRef]

1973

A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973).
[CrossRef]

1960

J. V. Wehausen and E. V. Laitone, “Surface waves,” Encyclopedia Phys. 9, 446–476 (1960).

1957

R. C. Thorne, “The asymptotic expansion of Legendre functions of large degree and order,” Phil. Trans. R. Soc. A 249, 597–620 (1957).
[CrossRef]

1949

D. Sinclair and V. K. la Mer, “Light scattering as a measure of particle size in aerosols,” Chem. Rev. 44, 245–267 (1949).
[CrossRef]

1908

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Anand, S.

S. Anand, A. Engelbrecht, and D. McGloin, “Optically written optofluidic ice channels,” J. Opt. 13, 044005 (2011).
[CrossRef]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767–784 (2001).
[CrossRef]

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef]

Arquis, E.

H. Chrabi, D. Lasseux, R. Wunenburger, E. Arquis, and J.-P. Delville, “Optohydrodynamics of soft fluid interfaces: optical and viscous nonlinear effects,” Eur. Phys. J. E 32, 43–52 (2010).
[CrossRef]

Ash, P.

A. D. Ward, M. Berry, P. Ash, D. Woods, and C. D. Bain, “The polymerisation of emulsion droplets deformed using laser tweezers to create microscopic polymer particles,” Central Laser Facility Ann. Rep. 2006/2007 Section 7 (2007), pp. 199–201.

Ashkin, A.

A. Ashkin and J. M. Dziedzic, “Radiation pressure on a free liquid surface,” Phys. Rev. Lett. 30, 139–142 (1973).
[CrossRef]

Bain, C. D.

D. A. Woods, C. D. Mellor, J. M. Taylor, C. D. Bain, and A. Ward, “Nanofluidic networks created and controlled by light,” Soft Matter 7, 2517–2520 (2011).
[CrossRef]

A. D. Ward, M. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. 2006, 4515–4517 (2006).
[CrossRef]

A. D. Ward, M. Berry, P. Ash, D. Woods, and C. D. Bain, “The polymerisation of emulsion droplets deformed using laser tweezers to create microscopic polymer particles,” Central Laser Facility Ann. Rep. 2006/2007 Section 7 (2007), pp. 199–201.

Baroud, C. N.

M. L. Cordero, E. Verneuil, F. Gallaire, and C. N. Baroud, “Time-resolved temperature rise in a thin liquid film due to laser absorption,” Phys. Rev. E 79, 011201 (2009).
[CrossRef]

C. N. Baroud, M. Robert de Saint Vincent, and J.-P. Delville, “An optical toolbox for total control of droplet microfluidics,” Lab on a Chip 7, 1029–1033 (2007).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Bellouard, Y.

Berry, M.

A. D. Ward, M. Berry, C. D. Mellor, and C. D. Bain, “Optical sculpture: controlled deformation of emulsion droplets with ultralow interfacial tensions using optical tweezers,” Chem. Commun. 2006, 4515–4517 (2006).
[CrossRef]

A. D. Ward, M. Berry, P. Ash, D. Woods, and C. D. Bain, “The polymerisation of emulsion droplets deformed using laser tweezers to create microscopic polymer particles,” Central Laser Facility Ann. Rep. 2006/2007 Section 7 (2007), pp. 199–201.

Birkeland, O. J.

O. J. Birkeland and I. Brevik, “Nonlinear laser-induced deformations of liquid–liquid interfaces: an optical fiber model,” Phys. Rev. E 78, 066314 (2008).

Brasselet, E.

R. Wunenburger, B. Issenmann, E. Brasselet, C. Loussert, V. Hourtane, and J.-P. Delville, “Fluid flows driven by light scattering,” J. Fluid Mech. 666, 273–307 (2011).
[CrossRef]

Brevik, I.

S. Å. Ellingsen and I. Brevik, “Electrostrictive counterforce on fluid microdroplet in short laser pulse,” Opt. Lett. 37, 1928–1930 (2012).
[CrossRef]

S. Å. Ellingsen and I. Brevik, “Electrostrictive fluid pressure from a laser beam,” Phys. Fluids 23, 096101 (2011).
[CrossRef]

O. J. Birkeland and I. Brevik, “Nonlinear laser-induced deformations of liquid–liquid interfaces: an optical fiber model,” Phys. Rev. E 78, 066314 (2008).

A. Hallanger, I. Brevik, S. Haaland, and R. Sollie, “Nonlinear deformations of liquid–liquid interfaces induced by electromagnetic radiation,” Phys. Rev. E 71, 056601 (2005).
[CrossRef]

I. Brevik and R. Kluge, “Oscillations of a water droplet illuminated by a linearly polarized laser pulse,” J. Opt. Soc. Am. B 16, 976–985 (1999).
[CrossRef]

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
[CrossRef]

Casner, A.

A. Casner and J.-P. Delville, “Laser-induced hydrodynamic instability of fluid interfaces,” Phys. Rev. Lett. 90, 144503 (2003).

Chan, C. T.

J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5, 531–534 (2011), and supplementary information.
[CrossRef]

Chang, R. K.

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and Q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[CrossRef]

J.-Z. Zhang and R. K. Chang, “Shape distortion of a single water droplet by laser-induced electrostriction,” Opt. Lett. 13, 916–918 (1988).
[CrossRef]

Chen, G.

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Figures (7)

Fig. 1.
Fig. 1.

Setup considered. An initially spherical particle of fluid 1 enclosed within a fluid 2 is illuminated by a Gaussian beam so that the particle’s center is at the beam focus.

Fig. 2.
Fig. 2.

Critical damping curves for the three cases: (1) a droplet in air, (2) an air bubble in liquid, and (3) the two-fluid example of oil-in water where μ2/μ1=1.2 and ρ2/ρ1=0.32 were used. The abscissa is A1 for cases (1) and (3), and A2 for case (2). Shaded areas denote underdamped modes. The interval of l values that contribute significantly to the force density in Eq. (5), 2lα, are indicated (slanted and horizontal dashed lines) for air–water and oil–water, respectively, when λ0=1064nm.

Fig. 3.
Fig. 3.

Absolute square of electric field |Ew|2 inside droplets with lower and higher dielectric contrast. The droplet radius in this example is a=2μm. Beam waist at focus in (a) and (b) is 1.5 μm (κ=11.8), and infinite in (c) and (d). Vacuum wavelength is λ0=1064nm, so α=15.7 in all cases. Contours are at the same levels in (a) and (b) (15 contours between 0 and 2.5E02; white areas: E2>2.5E02), and in (c) and (d) (15 contours between 0 and 6E02; white areas: E2>6E02). Peak values are 1.43E02 (a), 4.35E02 (b), 2.07E02 (c), and 15.75E02 (d).

Fig. 4.
Fig. 4.

Approximate static shapes for the same configurations as in Fig. 3. The laser power in (a) is 7 mW (Peff=6.8mW), and the effective power in (c) is 4 times that of (a). The laser intensity in (d) is 6Wμm2 (Peff=75W) and the effective power in (b) is 1/3 that in (a) (25 W). Droplet radius is a=2μm in all examples.

Fig. 5.
Fig. 5.

Approximate “lemon” shape of oil-emulsion with very narrow laser beam. We use λ=385nm as used in [25,26], radius a=2.5μm (α=54.3), and waist w0=0.4μm (κ=8.7) so κ2/α=1.4, thus somewhat beyond the scope of the theory.

Fig. 6.
Fig. 6.

Bubble of air in water, for the same cases as considered in Figs. 3 and 4: a narrow beam (a)–(c) and a wide beam (d)–(f). Field intensity contour plots (a) and (d) are scaled the same as corresponding plots in Fig. 3 for direct comparison. Figures (b) and (e) show deformations using the same laser powers as used for water droplets in (b) and (d), respectively. Panels (c) and (f) show the same deformations as panels (b) and (e), respectively, but with power boosted by the indicated factor.

Fig. 7.
Fig. 7.

Dynamics of deformation for a water droplet, comparing the cases of static beam and a short pulse. (a) Water droplet in wide beam, as in Fig. 4(d), but with slightly higher light intensity, I0=10Wμm2. Static beam (upper row) compared to beam of duration 0.25 μs (lower row). (b) Time development of droplet elongation for the above shapes, illustrated by r(θ=0,t)/a (rear of droplet, plotted above abscissa) and r(θ=π,t)/a (front of droplet where light enters, plotted below abscissa). Static beam turned on at t=0 (solid) and pulse of duration t0=0.25μs (dashed) are shown. Panel (c) shows the same as (b) but for the overdamped case of an oil emulsion in a narrow beam, same setup as Fig. 4(a), but with power boosted to 12 mW, for static beam switched on at t=0 (solid) and a pulse switched off again at t0=10ms (dashed). Panels (d) and (e) show the time development of the four lowest deformation modes, as given in Eq. (33a), for a static beam switched on at t=0 (d) and a pulse switched off again at t0=0.25μs. The legend in panel (e) is for both panels (d) and (e).

Tables (1)

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Table 1. Qualitative Shapes of Fluid Particles Under Illumination by Narrow and Wide Laser Beamsa

Equations (86)

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f=12ϵ0E2n2+12[E2ρ(n2ρ)T]+n21c2t(E×H).
t1.4nsa1μm,liquid droplett5.9nsa1μm,gas bubble.
T=ED+HB12(E·D+H·B)1,
σ(Ω)=σrr=Trr(r=a+)Trr(r=a).
σ(θ)=ε0n224(n¯21)(n¯2|Erw|2+|Eθw|2+|Eϕw|2).
σ(θ)=l=0σlPl(cosθ).
σl=12(2l+1)0πdθsinθσ(θ)Pl(cosθ).
α=k2a=2πaλ2=n2ωac,
r(θ,t)=a+l=2hl(t)Pl(cosθ).
hl()=σla2γ1l2+l2,
hl(t)hl()=1(μlγlsinγlt+cosγlt)eμlt,ωl>μl
hl(t)hl()=1(μlΓlsinhΓlt+coshΓlt)eμlt,ωl>μl
γl=ωl2μl2;Γl=μl2ωl2,
ωl2=γlΔρa3(l2+l2),
μl=ν1a2(2l2l1)1+μ2μ1(l+1)(l+2)l(l1)1+ρ2ρ1ll+1.
A1>(2l2l1)2l(l2+l2),l2,
A1=aγΔρν12,
l>112{A+236+24A+A2cos[13arctan63A648+441A+136A2+4A3216+54A+36A2+A3]}{A/4,whenA11,whenA1,
A1=Ga1Eo,
Ga1=ga3ν12,Eo=Δρa2gγ.
A1waterair=72.5·a/1μm.
λ08πn2Δρν2γ,
λ0350nm,
A2>(2l+1)2(l+1)4(l+2)l5(l1),l2,
A2=aγΔρν22.
A2<82.23.
A1oilwater0.00050·a/1μm
ExiE0ψ0*eikz;Eyi=iExi;
Ezi2Q*kw02(x+iy)Exi;
Hiin2μ0cEi
ψ0=iQeiQ(x2+y2)/w02;Q=1i+2z/kw02.
κ=kw0.
κ2α,
λ0w02πn2w0a.
Erw=E0ϱ2l=1m=lll(l+1)c˜lmψl(n¯αϱ)Ylm(Ω),
Eθw=αE0ϱl=1m=ll[n¯c˜lmψl(n¯αϱ)θYlm(Ω)d˜lmn2mψl(n¯αϱ)Ylm(Ω)sinθ],
Eϕw=iαE0ϱl=1m=ll[mn¯c˜lmψl(n¯αϱ)Ylm(Ω)sinθd˜lmn2ψl(n¯αϱ)θYlm(Ω)],
c˜lm=iA˜lm[n¯2ψl(n¯α)ξl(1)(α)n¯ψl(n¯α)ξl(1)]1,
d˜lm=iB˜lm[ψl(n¯α)ξl(1)(α)n¯ψl(n¯α)ξl(1)]1.
A˜lmB˜lm=1l(l+1)ψl(α)Eri/E0Hri/H0Ylm*(Ω)dΩ,
EriE0=Exisinθeiϕ+EzicosθE0=sinθeiϕExE0[12iακ2cosθ+]=κ4sinθeiϕ(κ2+2iαcosθ)2exp[iακ2cosθα2(1+cos2θ)κ2+2iαcosθ].
Ylm(Ω)=2l+14π(lm)!(l+m)!Plm(cosθ)eimϕ
02πdϕeiϕeimϕ=2πδm1.
A˜lm=(2l+1)πδm1[l(l+1)]3/2ψl(α)Φl(α,κ);
B˜lm=in2A˜lm,
Φl(α,κ)=11duκ41u2Pl1(u)(κ2+2iαu)2exp[iακ2uα2(1+u2)κ2+2iαu],
limκΦl(α,κ)=2il+1α2l(l+1)ψl(α),
σl=ε0n22E02(n¯21)32m=1n=1{cmcn*ψm(n¯α)ψn(n¯α)Ilmn+α2[cmcn*ψm(n¯α)ψn(n¯α)+dmdn*ψm(n¯α)ψn(n¯α)]Mlmn+iα2[cmdn*ψm(n¯α)ψn(n¯α)dmcn*ψm(n¯α)ψn(n¯α)]Nlmn},
Ilmn=(2l+1)(2m+1)(2n+1)m(m+1)n(n+1)11duPl(u)Pm1(u)Pn1(u),
Mlmn=(2l+1)(2m+1)(2n+1)[m(m+1)n(n+1)]211duPl(u)[(1u2)Pm1(u)Pn1(u)+Pm1(u)Pn1(u)1u2],
Nlmn=(2l+1)(2m+1)(2n+1)[m(m+1)n(n+1)]211duPl(u)Pm1(u)Pn1(u)
cl=Φl(α,κ)ψl(α)1n¯ψl(n¯α)ξl(1)(α)ψl(n¯α)ξl(1),
dl=Φl(α,κ)ψl(α)1ψl(n¯α)ξl(1)(α)n¯ψl(n¯α)ξl(1).
I=Sz=E×Hz=12Re{E×H*}z=n2|Ex|2μ0c=n2|Q|2E02μ0ce2|Q|2(x2+y2)/w02I0|Q|2e2|Q|2(x2+y2)/w02,
P=12πw02I0=12πε0cn2w02E02.
σl=n2I0(n¯21)32cm=1n=1{}=n2P(n¯21)16πcw02m=1n=1{}.
Peff=πI0w022(1e2a2/w02).
Peffκ=πa2I0.
Δr(Ffront+Fback)12πγ=AP,
hl(t)hl()=1(μlγlsinγlt+cosγlt)eμlt+Θ(tt0){1[μlγlsinγl(tt0)+cosγl(tt0)]eμl(tt0)},ωl>μl
hl(t)hl()=1(μlΓlsinhΓlt+coshΓlt)eμlt+Θ(tt0){1[μlΓlsinhΓl(tt0)+coshΓl(tt0)]eμl(tt0)},ωl>μl
τldroplet=1μldroplet=4.0μs(2l2l1),
τlemulsion=(μlΓl)12μlωl2=4Δρaν¯γ1lf(l)=0.012slf(l),
(I,M,N)lmn=S^lmn(i,m,n)lmnlmn,
S^lmn=l=0lm=1mn=1n(ll)(mm)(nn)×(l+l12l)(m+m12m)(n+n12n)×e(l+l)e(m+m)e(n+n),
Λ=l+m+n,Λ=l+m+n,
e(n)={1,nis even0,nis odd,o(n)={0,nis even1,nis odd.
ilmnlmn=2Λ+2e(Λ)(Λ21)1,
mlmnlmn=2Λ+1e(Λ)mn[mnΛ+12mnmn1Λ1+(m1)(n1)Λ3],
nlmnlmn=2Λo(Λ)lmn(Λ22Λ)1.
φint=l=0ClPlm(cosθ)eimϕϱl,φext=l=0DlPlm(cosθ)eimϕϱl1.
DSDt=S˙+(v·)S=0,
φ˙|r=a=vr|r=a,
Dl=ll+1Cl.
E¯mech=ρ1intd3x(φint)2¯+ρ2extd3x(φext)2¯.
E¯mechint=πρ1al=0l=0m=llm=llClmClm*¯×01dϱϱl+l02πdϕei(mm)ϕ×11dx{(ll+mm1x2)Plm(x)Plm(x)+(1x2)[xPlm(x)][xPlm(x)]}.
E¯mechint=4πρ1al=0m=lll|Clm|2¯2l+1(l+m)!(lm)!.
E¯mech=4πal=0m=lll|Clm|2¯2l+1(l+m)!(lm)!(ρ1+lρ2l+1).
E˙¯mech=μ1a2dΩrvint2¯|ϱ=1+μ2a2dΩrvext2¯|ϱ=1=8πa|Clm|2¯l=0m=lll(l+m)!(lm)![μ1l(l1)+μ2(l+1)(l+2)].
μlm=12|E˙¯mech,lmE¯mech,lm|,
02πdϕei(mm)ϕ=2πδmm11dxPlm(x)Plm(x)=22l+1(l+m)!(lm)!δll.
Illm11dxf(x)11dx{m21x2Plm(x)Plm(x)+(1x2)[xPlm(x)][xPlm(x)]}.
[(1x2)x22xx+l(l+1)m21x]Plm(x)=0
(xPlm)(xPlm)=x2(PlmPlm)Plm(x2Plm)Plm(x2Plm),
f(x)=12{x(1x2)xPlmPlm+[l(l+1)+l(l+1)]PlmPlm}.
Illm=2l(l+1)2l+1(l+m)!(lm)!δll.

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